Fractions allow us to understand the division of a whole. We first learn about them early in our school days as they are part of mathematics that we frequently use in everyday life.

Whenever you cut up a pizza or a cake, you use fractions. If you want to cook something in the oven for an hour and need to turn it midway, you use fractions.

Fractions are an important mathematical tool that allow us to divide the world and understand it in its constituent parts.

So what exactly are they?

A fraction represents a part of a whole or several equal parts of a whole. There are two parts to a fraction, one tells us how many parts the whole has been divided into, and the other tells us how many of those parts we have.

Fractions are an important way of understanding non-integers, numbers that are not whole numbers. A fraction can also be written as a decimal or as a percent.

We will explore that in detail later, but let's start by looking at some terminology relating to fractions.

Each of the following terms relates to fractions. They are important for understanding how fractions work and for being able to talk about fractions in a formal context, such as an exam.

Numerator

The numerator is the top number of a fraction. It tells us how many parts of a whole we have.

For example, if we have a pizza cut into eight equal parts and only three slices are left, then we have 3/8 of the pizza. The numerator in this instance is 3, and it tells us that three parts of the total eight remain.

Denominator

The denominator is the bottom number of a fraction. It tells us how many equal parts a whole has been split into.

Let's use the same example as earlier.

If we have a pizza and cut it into eight equal parts, we start with 8/8 as the pizza has been split into eight and all parts eight remain. If five parts are eaten, we are left with 3/8. The denominator is eight because even though some slices are gone, eight is still the number the whole was divided into.

Quotient

The quotient is the answer you get when you divide two numbers. For example:

  • 10 ÷ 2 = 5
  • 5 is quotient

When it comes to fractions, the quotient is the number obtained by simplifying a fraction by dividing the numerator by the denominator.

For example:

  • 210
  • 2 ÷ 10 = 0.2
  • Therefore, 0.2 is the decimal form of 210

Divisor

The divisor is the number you use to divide another. For example:

  • 6 ÷ 3 = 2
  • 3 is the divisor because it is the number we have used to split 6.

Divisors are useful for fractions because the quotient of a division may not equal a whole number, and we can write this as a fraction. For example:

  • 100 ÷ 3 = 33
    1 / 3

We can also use fractions to express divisions. So let's use our earlier example:

  • 6 ÷ 3 = 2
  • 6 ÷ 2 = 3
  • Therefore...
  • 36 = 12
  • 26 = 13

There are several different types of fractions. Here, we will look at some of the main ones you will likely see.

Proper fraction

A proper fraction is a fraction in which the numerator is smaller than its denominator.

For example:

  • 23 is a proper fraction because two (the numerator) is smaller than three (the denominator).

It is called a proper fraction because it only represents fractional values that are less than one whole, and there are no integers involved (which there are with improper fractions and mixed numbers).

Improper fraction

An improper fraction is a fraction in which the numerator is equal to or greater than its denominator.

For example:

  • 88 is an improper fraction because the numerator and denominator are both eight. It could also be written as 1.
  • 108 is also an improper fraction because ten (the numerator) is greater than eight (the denominator). It could also be written as 1 28.

Mixed number

A mixed number (or a mixed fraction) is a number comprised of both a whole number and a fraction.

For example:

  • 1 13 is a mixed number.

Unit fraction

A unit fraction is any fraction in which the numerator is 1.

For example:

  • 13, 18, and 1200 are all unit fractions.

Simplified fractions refer to two fractions of the same value. You can simplify fractions to find the simplest form of a fraction and to make it easier to compare fraction sizes.

For example:

  • 48, 36, and 24, are all the equivalent fraction. Each fraction represents 12 as the denominator is double the value of the numerator.

To simplify a fraction, you need to:

  • find the highest common factor of both the numerator and the denominator.
  • So, for 36, the highest common factor is three because three is the highest number that both three and six can be divided by.
  • You then divide both the numerator and the denominator by the highest common factor.
  • So, for 36, you would be left with 12. This is because 3 ÷ 3 = 1 and 6 ÷ 3 = 2.

If there isn't a factor that both numbers can divide by, then the fraction is already in its simplest form.

If you want to add two fractions or more together and they all have the same denominator, you simply add the numerators together and leave the denominator as it is.

For example:

  • 16 + 36 = 46
  • This is because 1 + 3 = 4. This gives you your numerator value.
  • The denominator stays the same because it is equal in both fractions.

If you want to add fractions together and the denominators are not equal, you first need to make your denominators equal. You do this by finding the lowest common multiple of the numbers.

For example:

  • 24 + 16 = ?
  • We need to make the denominators (4 and 6) equal.
  • What is the lowest number that both 4 and 6 go into?
  • The answer is 12.
  • 4 x 3 = 12.
  • 6 x 2 = 12.
  • We then need to multiply our numerators by the same number as our denominators.
  • We multiplied 4 by 3. So now we need to multiply 2 by 3 to get our first numerator.
  • 2 x 3 = 6.
  • We multiplied 6 by 2. So now we need to multiply 1 by 2.
  • 1 x 2 = 2.
  • So our fractions are now: 612 + 212.
  • We then follow the process we outlined earlier and add our numerators to get...
  • 812
  • So, 24 + 16 = 812

If you want to subtract fractions from one another and they have the same denominator, you subtract the numerators and leave the denominator as it is.

For example:

  • 56 - 26 = 36
  • This is because 5 - 2 = 3.
  • So our numerator value is 3 and our denominator value stays the same.

Subtracting fractions with different denominators requires you first to make your denominators equal. You do this by following the process we outlined in the previous section. Then, once the denominators are equal, you subtract the numerators from one another.

If you want to multiply fractions, the denominators don't have to be equal. You multiply the numerators together and then the denominators.

For example:

  • 12 x 25 = ?
  • First, multiply your numerators.
  • 1 x 2 = 2.
  • Next, multiply your denominators.
  • 2 x 5 = 10.
  • Therefore...
  • 12 x 25 = 210

You can then simplify 210 by dividing the numerator and the denominator 2.

  • So 210 = 15

Dividing fractions is a little bit more complicated, but it can still be done quickly once you know what you are doing.

Let's say you are given the question:

  • 12 ÷ 16 = ?

Here is what you do:

  • You begin by turning the second fraction into a reciprocal fraction. This means you turn it upside down. So 16 becomes 61.
  • Then, you multiply the numerator of the reciprocal fraction with the numerator of the first fraction. So 1 x 6 = 6.
  • Then you do the same with the denominators. 2 x 1 = 2.
  • So now we are left with 62.
  • 62 can be simplified to 3, which is our answer.
  • 12 ÷ 16 = 3

Half can be written as:

  • 12
  • 50%
  • 0.5

Each version is just as correct as the other, though different contexts may require that you choose one in particular.

To convert a fraction to a decimal, you divide the numerator by the denominator. For example:

  • 12
  • 1 ÷ 2 = 0.5
  • 12 = 0.5

To convert a decimal into a fraction, you do the following:

  • Let's say you want to convert 0.75 into a fraction. You begin by putting the decimal into a fraction over 1.
  • 0.751
  • You then multiply the numerator and the denominator by 10 for every number after the decimal point. So multiply by 10 for one number, 100 for two, etc.
  • So we get: 0.75 x 100 = 75 and 1 x 100 = 100
  • This gives us 75100
  • Then simplify the fraction to get 34
  • 0.75 = 34

To convert a fraction to a percentage, we divide the numerator by the denominator and then multiply the answer by 100.

For example, let's say we want to convert 38 to a percentage:

  • 3 ÷ 8 = 0.375
  • 0.375 x 100 = 37.5
  • 38 = 37.5%

To convert a percentage to a fraction, we first convert the percentage to a decimal (divide it by 100) and then use the same process of converting decimals to fractions.

For example, let's say we want to convert 80% to a fraction.

  • Turn 80% into a decimal by dividing by 100.
  • 80100 = 0.8.
  • Write the decimal over 1.
  • 0.81
  • Multiply the numerator and denominator by 10 for every number after the decimal point.
  • 0.8 x 10 = 8
  • 1 x 10 =10
  • This gives us 810
  • Simplify the fraction to
  • 45
  • 80% = 45